Acronym  bipach 
Name  biprismatocubic honeycomb 
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This isogonal honeycomb cannot be made uniform, i.e. having all 3 edge types at the same size.
The incidence matrix below shows that the vertex figure is a triapiculated triangular pyramid.
Note that the below provided value of c (obtained from the zero height requirement of the tegum sum), when inserted into the formula of the dihedral angle of the lacing edge of the recta results in arccos(1/[4c^{2}/(ba)^{2}1]) = arccos(1/2) = 60° = 360°/6, i.e. the below provided seemingly huge number of 6 trapezoprisms around the cedges indeed is exact here. Note moreover that the calculated value does no longer depend on the chosen ratio of a:b !
Sure, asking that the lacing edges of the pyramid are outside to the burried pseudo edges of the medial layer triangle, might give some further restriction to the a:b ratio, which so far has not been evaluated.
Squares and rectangles here are coplanar. In fact, the cubes are situated in the vertex positions of a body centered cubic lattice, and the square prisms are situated in the edge positions of the 2 interwoven inscribed cubic honeycombs. The recta then simply interconnect those.
Incidence matrix according to Dynkin symbol
ab4oo3oo4ba&#zc (N → ∞) → height = 0 a < b c = ab sqrt(3)/2 o.4o.3o.3o. &  8N  3 3 1  3 6 6  1 3 6 ++++ a. .. .. .. &  2  12N * *  2 2 1  1 2 2 .. .. .. b. &  2  * 12N *  0 2 1  0 1 2 oo4oo3oo4oo&#c  2  * * 4N  0 0 6  0 0 6 ++++ a.4o. .. .. &  4  4 0 0  6N * *  1 1 0 square a. .. .. b. &  4  2 2 0  * 12N *  0 1 1 rectangle ab .. .. ..&#c &  4  1 1 2  * * 12N  0 0 2 trapezium ++++ a.4o.3o. .. &  8  12 0 0  6 0 0  N * * cube a.4o. .. b. &  8  8 4 0  2 4 0  * 3N * long square prism ab .. .. ba&#c  8  4 4 4  0 2 4  * * 6N recta
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